Modeling Graphs Beyond Hyperbolic: Graph Neural Networks in Symmetric Positive Definite Matrices
This work addresses the need for more sophisticated geometric embedding spaces in graph representation learning, offering a novel approach for handling complex graph data, though it is incremental as it builds on existing graph neural network architectures.
The authors tackled the problem of representing complex real-world graphs with multiple geometric and topological features by developing graph neural networks in symmetric positive definite (SPD) matrices, resulting in substantial performance improvements over Euclidean and hyperbolic counterparts on node and graph classification tasks.
Recent research has shown that alignment between the structure of graph data and the geometry of an embedding space is crucial for learning high-quality representations of the data. The uniform geometry of Euclidean and hyperbolic spaces allows for representing graphs with uniform geometric and topological features, such as grids and hierarchies, with minimal distortion. However, real-world graph data is characterized by multiple types of geometric and topological features, necessitating more sophisticated geometric embedding spaces. In this work, we utilize the Riemannian symmetric space of symmetric positive definite matrices (SPD) to construct graph neural networks that can robustly handle complex graphs. To do this, we develop an innovative library that leverages the SPD gyrocalculus tools \cite{lopez2021gyroSPD} to implement the building blocks of five popular graph neural networks in SPD. Experimental results demonstrate that our graph neural networks in SPD substantially outperform their counterparts in Euclidean and hyperbolic spaces, as well as the Cartesian product thereof, on complex graphs for node and graph classification tasks. We release the library and datasets at \url{https://github.com/andyweizhao/SPD4GNNs}.